On Distance Function in Some New Analytic Bergman Type Spaces in C n Romi

We extend our previous sharp results on distances obtained for analytic Bergman type spaces in unit disk to some new analytic Bergman type spaces in higher dimensions in . Also, we study the same problem in anisotropic mixed norm spaces consisting of -harmonic functions on the unit polydisc of .


Introduction and Preliminaries
The goal of this paper is to add several new results for distances in analytic Bergman type and Bergman type nharmonic function spaces of functions of several variables.
It turns out that some of our distance theorem, which we proved before for the case of unit disk, are valid under certain natural conditions in various domains and various Bergman type analytic spaces.Namely, we look at analytic Bergman spaces in minimal ball and so-called analytic -Bergman spaces and in analytic Bergman-Orlicz spaces.These analytic spaces act as direct extensions of well-known analytic Bergman spaces in the unit disk.These analytic spaces are relatively new and we will include some basic facts on them in our paper; they will also be needed for proofs of our assertions partially.We will turn also to polydisk and -harmonic function spaces on them providing also a sharp result there.Note in less general case, namely, in analytic Bergman type spaces in polydisk, this result is already known (see [1,2]).
Our intention in this paper is the same as our previous papers on this topic; namely, we collect some facts from earlier investigation concerning Bergman projection and Bergman kernel and use them for our purposes in estimates of dist  (, X) function (distance function).
Following our previous paper [1,2], we easily see that, to obtain a sharp result for distance function, we only need several tools and the scheme here is as follows.First we need an embedding of our quazinormed analytic space (in any domain) into another one ( ⊂ ), this immediately poses a problem of dist  (, ) = inf ∈ ‖ − ‖  for all  ∈  \ , and then we need the Bergman reproducing formula for all  functions from  space.Then, finally we use the boundedness of Bergman type projections, with |(, )| positive kernel acting from  to , together with Forelli-Rudin type sharp estimates of Bergman kernel.These three tools were used in general Siegel domain of second type, polydisk, and unit ball in [1][2][3][4] (see also various references there).We continue to use these tools providing new sharp (and not sharp) results in various spaces of analytic functions in this paper.
First we provide a known result in the unit disk with complete proof taken from our previous paper ( [1,2]).In the unit disk case all arguments here are short and transparent and are based on several tools like Forelli-Rudin type estimate and estimates for Bergman type projections with positive Bergman kernel, and then we will see arguing similarly as in unit disk; we will complete the proof of more complicated cases easily.The complete formulation of our theorems will be given, but only sketches of proofs will be added and details of proofs of higher dimensional cases will be left to readers.
By  2 we denote the volume measure on U  and by   we denote the normalized Lebesgue measure on   .Let (U  ) be the space of all holomorphic functions on U  .
Throughout the paper, we write  (sometimes with indexes) to denote a positive constant which might be different at each occurrence (even in a chain of inequalities) but is independent of the functions or variables being discussed.
The notation  ≍  means that there is a positive constant , such that / ≤  ≤ .We will write for two expressions  ≲  if there is a positive constant  such that  < .
The Hardy spaces, denoted by   (U  ) (0 <  ≤ ∞), are defined by   (U  ) = { ∈ (U  ): sup 0≤<1   (, ) < ∞}, where As usual, we denote by ⃗  the vector ( 1 , . . .,   ).For   > −1,  = 1, . . ., , 0 <  < ∞, recall that the weighted Bergman space   ⃗  (U  ) consists of all holomorphic functions on the polydisk such that When  1 = ⋅ ⋅ ⋅ =   = , then we use notation    (U  ).If  is -harmonic (harmonic by each variable), then as usual Also, we denote by the space of all functions f holomorphic in B n , satisfying the condition where ] is the Lebesgue measure in C  ≡ R 2 (see [6]).Further, for a complex number  with Re  > −1, put spaces are well known in literature as so-called growth spaces (see, e.g., [5]).It can be shown easily that these spaces are Banach spaces.These spaces are playing a vital role in this paper and are embedded in Bergman spaces    , for large enough .Hence the representation (10) with  large enough index depending on  for all functions from such classes is valid.The mentioned embedding is well known and almost obvious and we leave the proof of it to interested readers.This fact nevertheless is crucial for the proof of Theorems 18 and 20 below and actually serves as base of both proofs, along with well-known Forelli-Rudin estimates for unit ball (see [5]).
We have the following theorem.
We have the following lemmas.
We will need some facts from theory of Bergman-Orlicz spaces (see [10,11]).
Let now Ψ: [0,∞) → [0,∞) be a continuous nondecreasing function which vanishes and is continuous at 0. Given a probabilistic space (Ω, P), we define the Orlicz class  Ω (Ω, P) as the set of all (equivalence classes of) measurable functions  on Ω such that ∫ Ω Ψ(||/)P < ∞ for some 0 <  < ∞.We used to define the Morse-Transue space  Ψ (Ω, P) by and we also introduce the following set: These three sets are not vector spaces and do not coincide, but we have We also define the Luxembourg gauge on  Ψ (Ω, P) by This functional is homogeneous and is 0 if and only if  = 0 but it is not subadditive.We say that the two functions Ψ 1 and Ψ 2 as above are equivalent if there exists some constant  such that for any  large enough.Two equivalent functions define the same Orlicz class with equivalent Luxembourg functionals.
In order to define a good topology on  Ψ (Ω, P) and to get properties convenient for our purpose, we will assume that Ψ satisfies the following definition.Definition 7. Let 0 <  ≤ 1.One says that Ψ: [0, ∞) → [0, ∞) is a growth function of order  if it satisfies the two following conditions.
We will say that Ψ is a growth function if it is a growth function of order  for some 0 <  ≤ 1.
Sometimes we will use lower  index for our growth function below.
When we deal with spaces of holomorphic functions, it is very natural to require subharmonicity.Then we will assume that Ψ is such that Ψ(||) is subharmonic when  is holomorphic.We will refer to such a function as a subharmonic-preserving function.
Here are some examples of (concave) growth function that we may consider further.
For Ψ a subharmonic-preserving growth function and  > −1, the weighted Bergman-Orlicz space  Ψ  of the ball consists of those holomorphic functions on B  which belongs to the Orlicz space  Ψ (B  , ]  ), where, as usual, for any real number , ] is the normalizing constant so that ]  has unit total mass.
Further, we will denote by ‖ ⋅ ‖ ,Ψ the corresponding Luxembourg (quasi)norm and by ‖ ⋅ ‖ Ψ  the quantity ∫ B  Ψ(||)]  . Ψ  is metric space for the distance  ,Ψ or  Ψ  defined by, respectively,  ,Ψ (, ) = ‖ − ‖ ,Ψ and  Ψ  (, ) = ‖ − ‖ Ψ  .If Ψ() =   , then we recover the usual weighted Bergman space    .We have the following inclusions: whenever Ψ is a growth function of order .Note that the second inclusion which we see before leads to another inclusion, namely,  Ψ  , is embedded in  ∞ (++1)/ .This last inclusion will lead us to distance theorem.We showed a complete analogue of a theorem in the unit disk below.
It is important to mention that a linear operator  from for any The next proposition says that the point evaluation functionals are continuous on  Ψ  .
Proposition 8 (see [11]).Let  > −1 and let Ψ be a subharmonic-preserving growth function.For any  ∈ B  and any  ∈  Ψ  , one has The proof is the same as that of [10, Proposition 1.9] and so is omitted (still use the hypothesis that Ψ(||) is subharmonic).We easily deduce from this and the completeness of  Ψ the following result.
Corollary 9 (see [11]). Ψ  , endowed with  ,Ψ or  Ψ  , is a complete metric space.Some facts from theory of -harmonic function spaces will be needed by us.We will use the same notation for   (, ) for harmonic and analytic functions.
For  =  < ∞ the spaces ℎ(, , ) coincide with the well-known weighted Bergman spaces, while for  = ∞ they are known as growth spaces.
Reproducing integral formula Poisson-Bergman type will be presented in the following theorem.
Let  , be a linear operator Bergman type Theorem C (see [13]).We will also need basic facts from theory of Bergman spaces in minimal ball (see [14,15]).For the convenience of readers we will give some details about them.
We consider the domain B * in C  ,  ≥ 2, defined by where for  and  in C  .This is the unit ball of C  with respect to the norm The norm  :=  * / √ 2 was introduced by Hahn and Pflug in [16], where it was shown to be the smallest norm in C  that extends the Euclidean norm in R  under certain restrictions.The domain B * has since been studied by Hahn and Pflug [16], Mengotti and Youssfi [15], and Oeljeklaus and Youssfi [17].In particular, it is well known that the automorphism group of B * is compact and its identity component is where the  1 -action is diagonal and the (, R)-action is by matrix multiplication (see [17], e.g.).It is also well known that B * is a nonhomogeneous domain.Its singular boundary consists of all points  with  •  = 0, and the regular part of the boundary of B * consists of strictly pseudoconvex points.

Journal of Function Spaces
For each  > −1 we let V  denote the measure on B * , defined by where V denotes the normalized Lebesgue measure on B * .For all 0 <  < ∞ we define where H(B * ) is the space of all holomorphic functions on B * .Naturally, the spaces A   (B * ) are called weighted Bergman spaces of B * .
When  = 2, there exists an orthogonal projection from  2 (B * , V  ) onto A 2  (B * ).This will be called the weighted Bergman projection and is denoted by P ,B * .
It is well known that P ,B * is an integral operator on  2 (B * , V  ), namely, where is the weighted Bergman kernel.Here and A(, ) is the sum As usual, is the binomial coefficient and V  (B * ) is the weighted Lebesgue volume of B * .See [15,17] for these formulas and more information about the weighted Bergman kernels.Because of the infinite sum (, ) the formula for K ,B * is not really a closed form.As such, it is inconvenient for us to do estimates for P ,B * directly.Thus we employ a technique that was used in [15].More specifically, we relate the domain B * to the hypersurface M of the Euclidean unit ball in C +1 defined by and It was proved in [17] that there is an ( + 1, C)-invariant holomorphic form  on H.Moreover, this form is unique up to a multiplicative constant.In fact, after appropriate normalization, the restriction to H ∩ (C \ {0}) +1 of this form is given by Our norm estimates for the weighted Bergman projection P ,B * on   spaces will be based on the corresponding estimates on M. Therefore, we will consider the spaces    (M) consisting of measurable complex-valued functions  on M such that where We use    (M) to denote the subspace of all holomorphic functions in    (M).The weighted Bergman projection P ,M is then the orthogonal projection from  2  (M) onto  2  (M).Again, it is well known that P ,M is an integral operator on  2  (M) given by the formula where K ,M is the corresponding Bergman kernel.The starting point for our analysis is the following closed form of K ,M , which was obtained as Theorem 3.2 in [15].Lemma 11 (see [14]).The weighted Bergman kernel K ,M of  2  (M) is given by where  is a certain constant that depends on  and .
Let : B * → C be a measurable function.We define a function T on M by The operator T will also play a key role in our analysis.In particular, we need the following result which was obtained as Lemma 4.1 in [10].
Theorem B of [15] states that the operator with positive kernel P ,B * maps   (B * , | • | (−2)/2 V  ) boundedly onto A   (B * ) for all  > 1.We need to derive an improved version of the classical Forelli-Rudin integral estimates in the case of the minimal ball.

This proposition leads directly to embeddings between A ∞
and A   1 for some  and  1 as we had above for unit disk case (see [2]) and we showed that this repeating arguments of unit disk case leads to new theorem we formulate below, in next section, for Bergman spaces in minimal ball.

On Distances in 𝐾-Bergman Spaces, Large
Bergman-Orlicz Spaces in C  , and Bergman

Spaces in Minimal Ball
Based fully on preliminaries of previous section, the plan of this section is the following: first we formulate a result in the unit disk and then repeat arguments we provided in proof of that theorem in various situations in higher dimensions.Since all proofs are short, the repetitions of arguments which are needed in higher dimension will be omitted; only sketches will be given.
It is easy to note, based on previous section results, that the complete analogues of the embedding we just provided are valid also in Bergman spaces in minimal ball for harmonic function spaces in polydisk and in -Bergman spaces, and this is based completely on preliminaries we have in previous section.We leave this easy task to readers.This allows us to pose a distance problem in each space we consider in this paper.
In the following theorem (see [1]) we calculate distances from a weighted Bloch class to Bergman spaces for  ≤ 1.It is easy to note that each argument below is also valid not only in unit disk but based on preliminaries in the previous section in Bergman spaces in minimal ball and polydisk and -Bergman spaces and Bergman-Orlicz spaces.So this theorem is very typical for us though it is known.
The following theorem is a version of Theorem 15 for the case  > 1.
Theorem 16.Let  > 1,  < 0,  ≤  − 1/,  > (−1 − )/, and  > −1−.Let  ∈  ∞ − .Then the following are equivalent: The proof of Theorem 16 is the same actually as the proof of Theorem 15.The only difference is the boundedness of Bergman type projection operator but with positive Bergman kernel.This fact will be used by us below heavily.
Indeed the close inspection of the proof of Theorem 15 shows that the proof of Theorem 16 is the same as the proof of Theorem 15. but here we will use (70) (see below) instead of (67).For  > 0,  > 1,  > 0,  > −1/ (see [18]) which follows immediately from Hölder's inequality and (68) (see [18]).We formulate one more theorem in this direction. where Note also that now even a more general theorem for all mixed norm spaces is valid, namely, for ℎ(, , ) spaces based on facts we provided already above concerning projection theorems.We leave this to interested readers.
The scheme of proofs of all high dimensional theorems of this section is the same.The main tools are (see [1,2]) the Bergman representation formula for larger space (for each  function from ,  ∈ ,  ⊂ ).Then we use the fact that ‖||‖  ≤ ‖‖  ,  ∈ , where || is a Bergman projection with positive Bergman kernel.Then we use Forelli-Rudin type estimate.As example we provide a complete proof of Theorem 17.For a parallel proof of other assertions we refer the reader to facts we provided above taken from mentioned papers and leave this procedure to readers.Now we return to the proof of Theorem 17.We follow arguments of unit disk case.

Proof of Theorem 17. Let sup 𝑟∈𝐼
) < ∞ for all   >  0 ,  = 1, . . ., , where  0 is large enough.This means that based on lemmas on -harmonic functions from the previous section that for large enough β , β > β0 ,  = 1, . . ., ,  The following theorem is based heavily on Remark 1, which we provided above, and Forelli-Rudin type estimates for unit ball (see [5] and Proposition 3) and some elementary arguments we see in proof of unit disk case.
The following theorem is based heavily on Remark 1, which we provided above, and Forelli-Rudin type estimates for unit ball (see [5] and Proposition 3) and some elementary arguments we see in proof of unit disk case.
We formulate complete analogue of Theorem 15 in Bergman-Orlicz spaces.